16 Anisotropic Extension of the Exchange Factor Transformation

16.1 Introduction

The exchange factor transformation described in Chapter Chapter 9 enables analytical treatment of multiple reflection-scattering processes in radiative transfer by transforming the exchange factor matrix \mathbf{F} into absorption and reflection-scattering matrices \mathbf{A} and \mathbf{R}. The method assumes isotropic reflection-scattering behavior, where the probability of reflection-scattering from element j is independent of the incident element i. This is represented by a column-constant single reflection-scattering matrix \mathbf{B}.

However, many physical systems exhibit anisotropic reflection-scattering, where the probability and directional distribution of reflection-scattering depend on the incident direction. This chapter presents an extension of the exchange factor transformation that accommodates anisotropic reflection-scattering while preserving all fundamental properties: non-negativity, energy conservation, and analytical solvability.

16.2 Anisotropic Reflection-Scattering Matrix

The anisotropic extension introduces a full matrix \mathbf{\Gamma} (Gamma) to characterize directional dependence of reflection-scattering:

\mathbf{\Gamma} \in \mathbb{R}^{N \times N}, \quad 0 \leq \mathbf{\Gamma}_{ij} \leq 1 \quad \forall \; ij \tag{16.1}

where \mathbf{\Gamma}_{ij} represents the relative probability of reflection-scattering from element j when radiation is incident from element i.

16.2.1 Physical Interpretation

\mathbf{\Gamma}_{ij} = 1: Maximum reflection-scattering from j due to incidence from i (isotropic limit)
\mathbf{\Gamma}_{ij} = 0: No reflection-scattering from j due to incidence from i (complete directional selectivity)
0 < \mathbf{\Gamma}_{ij} < 1: Partial anisotropic reflection-scattering with directional preference

When \mathbf{\Gamma} = \mathbf{1} (all ones), the anisotropic formulation reduces exactly to the isotropic case, ensuring backward compatibility and providing a natural basis for proof by continuity arguments.

16.3 Construction of the Anisotropic Interaction Matrix

The anisotropic interaction-reflection-scattering matrix \mathbf{K} is constructed through a multi-step process that ensures physical consistency.

Step 1: Initial Reflection-Scattering Distribution

Define the initial reflection-scattering distribution matrix:

\mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j + \mathbf{\Gamma}_{ij} \cdot (1 - \mathbf{b}_j) \tag{16.2}

where \mathbf{b}_j is the reflection-scattering coefficient of element j. This constraint ensures that highly reflecting-scattering element must remain near isotropic while less reflecting-scattering elements can have more anisotropic freedom.

Each column j of \mathbf{\Gamma}_{ij} is scaled by 1-\mathbf{b}_j of the receiving element.

Step 2: Interaction-Reflection-Scattering (Pre-Normalization)

Compute the element-wise product of the exchange factor matrix and the initial reflection-scattering distribution:

\mathbf{K}_{\mathrm{init},ij} = \mathbf{F}_{ij} \cdot \mathbf{B}_{\mathrm{init},ij} = \mathbf{F}_{ij} \cdot \mathbf{\Gamma}_{ij} \cdot (1 - \mathbf{P}_{jj}) \tag{16.3}

This represents the probability that radiation emitted from element i interacts with element j and is then reflected-scattered, before normalization.

Step 3: Enforce Probability Conservation

Create a matrix that accounts for both reflection-scattering and absorption:

\mathbf{U}_{ij} = \mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \tag{16.4}

Each element either reflects-scatters (\mathbf{K}_{\mathrm{init}}) or is absorbed (\mathbf{F} \cdot \mathbf{P}).

Step 4: Compute Row Sums

Calculate the row sums:

\mathbf{s}_i = \sum_j \mathbf{U}_{ij} = \sum_j (\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}) \tag{16.5}

Expanding:

\mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{\Gamma}_{ij} \cdot (1-\mathbf{P}_{jj}) + \mathbf{P}_{jj}) \tag{16.6}

Step 5: Normalize to Row-Stochastic

Normalize the matrix to ensure row sums equal unity:

\mathbf{T}_{ij} = \frac{\mathbf{U}_{ij}}{\mathbf{s}_i} \tag{16.7}

This creates a row-stochastic matrix, ensuring that total probability is conserved for radiation emitted from each element i.

Step 6: Final Interaction-Reflection-Scattering Matrix

Extract the reflection-scattering component:

\mathbf{K}_{ij} = \mathbf{T}_{ij} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \tag{16.8}

Substituting from Eq. 16.7:

\mathbf{K}_{ij} = \frac{\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}}{\mathbf{s}_i} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \tag{16.9}

Simplifying:

\mathbf{K}_{ij} = \frac{\mathbf{K}_{\mathrm{init},ij}}{\mathbf{s}_i} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \left(1 - \frac{1}{\mathbf{s}_i}\right) \tag{16.10}

16.4 Properties of the Row Sum Normalization Factor

16.4.1 Bounds on s

From Eq. 16.6:

\mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{\Gamma}_{ij} \cdot (1-\mathbf{P}_{jj}) + \mathbf{P}_{jj})

Since \mathbf{F} is row-stochastic (\sum_j \mathbf{F}_{ij} = 1), we can write:

\mathbf{s}_i - 1 = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{\Gamma}_{ij} \cdot (1-\mathbf{P}_{jj}) + \mathbf{P}_{jj} - 1) \tag{16.11}

Simplifying:

\mathbf{s}_i - 1 = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{\Gamma}_{ij} - 1) \cdot (1-\mathbf{P}_{jj}) \tag{16.12}

Since \mathbf{\Gamma}_{ij} \leq 1, \mathbf{F}_{ij} \geq 0, and (1-\mathbf{P}_{jj}) \geq 0:

\mathbf{s}_i - 1 \leq 0 \implies \mathbf{s}_i \leq 1 \tag{16.13}

Furthermore, the lower bound \mathbf{P}_{jj} = 1-\mathbf{b}_j > 0 creates the lower bound S > 0, leading to the full bounds on S

0 < \mathbf{s}_i \leq 1

which prevents division by zero in equation (Eq. 16.9) and (Eq. 16.10)

16.4.2 Special Case: Γ = 1 (Isotropic)

When \mathbf{\Gamma} = \mathbf{1} (all ones), we have:

\mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot ((1-\mathbf{P}_{jj}) + \mathbf{P}_{jj}) = \sum_j \mathbf{F}_{ij} = 1

Therefore, the normalization does nothing, and:

\mathbf{K}_{ij} = \mathbf{F}_{ij} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} = \mathbf{F}_{ij} \cdot (1-\mathbf{P}_{jj})

This recovers the isotropic case exactly: \mathbf{K} = \mathbf{F} \circ \mathbf{B}, where \mathbf{B} is the column-constant matrix of reflection-scattering coefficients.

16.5 Subsequent Transformation

Once \mathbf{K} is constructed, the remainder of the exchange factor transformation proceeds almost identically to the isotropic case:

The steady-state path matrix is given by:

\mathbf{S}_\infty = (\mathbf{I} - \mathbf{K})^{-1} \mathbf{F} \tag{16.14}

The absorption and reflection-scattering matrices are given by the alternate formulation:

\mathbf{A} = \mathbf{P}(\mathbf{I}-\mathbf{K})^{-1}(\mathbf{F}-\mathbf{K}) \tag{16.15}

\mathbf{R} = \mathbf{P}(\mathbf{I}-\mathbf{K})^{-1}\mathbf{K} \tag{16.16}

The system matrice \mathbf{C} and \mathbf{D} are given by:

\mathbf{C} = \mathbf{I} - \mathbf{A}^T - \mathbf{R}^T \tag{16.17}

\mathbf{D} = \mathbf{I} - \mathbf{R}^T \tag{16.18}

The mixed boundary system is solved as:

\mathbf{M}\mathbf{j} = \mathbf{h} \tag{16.19}

where \mathbf{M} is assembled row-by-row from \mathbf{C} (for known sources) and \mathbf{D} (for known emissive powers), and \mathbf{j} is the total radiant power vector.

16.6 Advantages of the Anisotropic Extension

Physical Fidelity: Captures directional dependence of reflection-scattering, enabling more accurate modeling of real materials and surfaces.
Analytical Framework: Maintains the analytical structure of the exchange factor transformation, avoiding iterative Monte Carlo methods.
Backward Compatibility: Reduces exactly to the isotropic case when \mathbf{\Gamma} = \mathbf{1}, ensuring consistency with established methods.
Preserved Properties: Non-negativity and energy conservation are guaranteed (see chapter Chapter 19 for proofs).
Flexibility: Accommodates any anisotropic distribution within the constraint 0 \leq \mathbf{\Gamma}_{ij} \leq 1, providing maximum modeling flexibility.

16.7 Physical Applications

For surfaces with specular components, media with anisotropic scattering phase functions (e.g., Mie scattering, Henyey-Greenstein, i.e. (ij,k)-three-index dependencies) and surfaces with oriented microstructure (e.g., brushed metal, fibrous materials), more complex models are required.

16.8 Conclusion

The anisotropic extension of the exchange factor transformation provides a mathematically rigorous and physically meaningful framework for modeling directional reflection-scattering in radiative transfer. By introducing the anisotropy matrix \mathbf{\Gamma} and employing row-stochastic normalization, the method preserves all essential properties while significantly expanding the range of physical phenomena that can be accurately modeled.

# Anisotropic Extension of the Exchange Factor Transformation ## Introduction The exchange factor transformation described in Chapter @sec-exchange_fact_trans enables analytical treatment of multiple reflection-scattering processes in radiative transfer by transforming the exchange factor matrix $\mathbf{F}$ into absorption and reflection-scattering matrices $\mathbf{A}$ and $\mathbf{R}$. The method assumes isotropic reflection-scattering behavior, where the probability of reflection-scattering from element $j$ is independent of the incident element $i$. This is represented by a column-constant single reflection-scattering matrix $\mathbf{B}$. However, many physical systems exhibit anisotropic reflection-scattering, where the probability and directional distribution of reflection-scattering depend on the incident direction. This chapter presents an extension of the exchange factor transformation that accommodates anisotropic reflection-scattering while preserving all fundamental properties: non-negativity, energy conservation, and analytical solvability. ## Anisotropic Reflection-Scattering Matrix The anisotropic extension introduces a full matrix $\mathbf{\Gamma}$ (Gamma) to characterize directional dependence of reflection-scattering: $$ \mathbf{\Gamma} \in \mathbb{R}^{N \times N}, \quad 0 \leq \mathbf{\Gamma}_{ij} \leq 1 \quad \forall \; ij $$ {#eq-gamma-bounds} where $\mathbf{\Gamma}_{ij}$ represents the relative probability of reflection-scattering from element $j$ when radiation is incident from element $i$. ### Physical Interpretation - $\mathbf{\Gamma}_{ij} = 1$: Maximum reflection-scattering from $j$ due to incidence from $i$ (isotropic limit) - $\mathbf{\Gamma}_{ij} = 0$: No reflection-scattering from $j$ due to incidence from $i$ (complete directional selectivity) - $0 < \mathbf{\Gamma}_{ij} < 1$: Partial anisotropic reflection-scattering with directional preference When $\mathbf{\Gamma} = \mathbf{1}$ (all ones), the anisotropic formulation reduces exactly to the isotropic case, ensuring backward compatibility and providing a natural basis for proof by continuity arguments. ## Construction of the Anisotropic Interaction Matrix The anisotropic interaction-reflection-scattering matrix $\mathbf{K}$ is constructed through a multi-step process that ensures physical consistency. **Step 1:** Initial Reflection-Scattering Distribution Define the initial reflection-scattering distribution matrix: $$ \mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j + \mathbf{\Gamma}_{ij} \cdot (1 - \mathbf{b}_j) $$ {#eq-b-init} where $\mathbf{b}_j$ is the reflection-scattering coefficient of element $j$. This constraint ensures that highly reflecting-scattering element must remain near isotropic while less reflecting-scattering elements can have more anisotropic freedom. Each column $j$ of $\mathbf{\Gamma}_{ij}$ is scaled by $1-\mathbf{b}_j$ of the receiving element. **Step 2:** Interaction-Reflection-Scattering (Pre-Normalization) Compute the element-wise product of the exchange factor matrix and the initial reflection-scattering distribution: $$ \mathbf{K}_{\mathrm{init},ij} = \mathbf{F}_{ij} \cdot \mathbf{B}_{\mathrm{init},ij} = \mathbf{F}_{ij} \cdot \mathbf{\Gamma}_{ij} \cdot (1 - \mathbf{P}_{jj}) $$ {#eq-k-init} This represents the probability that radiation emitted from element $i$ interacts with element $j$ and is then reflected-scattered, before normalization. **Step 3:** Enforce Probability Conservation Create a matrix that accounts for both reflection-scattering and absorption: $$ \mathbf{U}_{ij} = \mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} $$ {#eq-enforce-unity} Each element either reflects-scatters ($\mathbf{K}_{\mathrm{init}}$) or is absorbed ($\mathbf{F} \cdot \mathbf{P}$). **Step 4:** Compute Row Sums Calculate the row sums: $$ \mathbf{s}_i = \sum_j \mathbf{U}_{ij} = \sum_j (\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}) $$ {#eq-row-sums} Expanding: $$ \mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{\Gamma}_{ij} \cdot (1-\mathbf{P}_{jj}) + \mathbf{P}_{jj}) $$ {#eq-row-sums-expanded} **Step 5:** Normalize to Row-Stochastic Normalize the matrix to ensure row sums equal unity: $$ \mathbf{T}_{ij} = \frac{\mathbf{U}_{ij}}{\mathbf{s}_i} $$ {#eq-normalize} This creates a row-stochastic matrix, ensuring that total probability is conserved for radiation emitted from each element $i$. **Step 6:** Final Interaction-Reflection-Scattering Matrix Extract the reflection-scattering component: $$ \mathbf{K}_{ij} = \mathbf{T}_{ij} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} $$ {#eq-k-final} Substituting from @eq-normalize: $$ \mathbf{K}_{ij} = \frac{\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}}{\mathbf{s}_i} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} $$ {#eq-k-explicit} Simplifying: $$ \mathbf{K}_{ij} = \frac{\mathbf{K}_{\mathrm{init},ij}}{\mathbf{s}_i} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \left(1 - \frac{1}{\mathbf{s}_i}\right) $$ {#eq-k-simplified} ## Properties of the Row Sum Normalization Factor ### Bounds on s From @eq-row-sums-expanded: $$ \mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{\Gamma}_{ij} \cdot (1-\mathbf{P}_{jj}) + \mathbf{P}_{jj}) $$ Since $\mathbf{F}$ is row-stochastic ($\sum_j \mathbf{F}_{ij} = 1$), we can write: $$ \mathbf{s}_i - 1 = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{\Gamma}_{ij} \cdot (1-\mathbf{P}_{jj}) + \mathbf{P}_{jj} - 1) $$ {#eq-s-minus-one} Simplifying: $$ \mathbf{s}_i - 1 = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{\Gamma}_{ij} - 1) \cdot (1-\mathbf{P}_{jj}) $$ {#eq-s-minus-one-simple} Since $\mathbf{\Gamma}_{ij} \leq 1$, $\mathbf{F}_{ij} \geq 0$, and $(1-\mathbf{P}_{jj}) \geq 0$: $$ \mathbf{s}_i - 1 \leq 0 \implies \mathbf{s}_i \leq 1 $$ {#eq-s-upper-bound} Furthermore, the lower bound $\mathbf{P}_{jj} = 1-\mathbf{b}_j > 0$ creates the lower bound $S > 0$, leading to the full bounds on $S$ $$ 0 < \mathbf{s}_i \leq 1 $$ which prevents division by zero in equation (@eq-k-explicit) and (@eq-k-simplified) ### Special Case: Γ = 1 (Isotropic) When $\mathbf{\Gamma} = \mathbf{1}$ (all ones), we have: $$ \mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot ((1-\mathbf{P}_{jj}) + \mathbf{P}_{jj}) = \sum_j \mathbf{F}_{ij} = 1 $$ Therefore, the normalization does nothing, and: $$ \mathbf{K}_{ij} = \mathbf{F}_{ij} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} = \mathbf{F}_{ij} \cdot (1-\mathbf{P}_{jj}) $$ This recovers the isotropic case exactly: $\mathbf{K} = \mathbf{F} \circ \mathbf{B}$, where $\mathbf{B}$ is the column-constant matrix of reflection-scattering coefficients. ## Subsequent Transformation Once $\mathbf{K}$ is constructed, the remainder of the exchange factor transformation proceeds almost identically to the isotropic case: The steady-state path matrix is given by: $$ \mathbf{S}_\infty = (\mathbf{I} - \mathbf{K})^{-1} \mathbf{F} $$ {#eq-s-infinity} The absorption and reflection-scattering matrices are given by the alternate formulation: $$ \mathbf{A} = \mathbf{P}(\mathbf{I}-\mathbf{K})^{-1}(\mathbf{F}-\mathbf{K}) $$ {#eq-a-matrix} $$ \mathbf{R} = \mathbf{P}(\mathbf{I}-\mathbf{K})^{-1}\mathbf{K} $$ {#eq-r-matrix} The system matrice $\mathbf{C}$ and $\mathbf{D}$ are given by: $$ \mathbf{C} = \mathbf{I} - \mathbf{A}^T - \mathbf{R}^T $$ {#eq-c-matrix} $$ \mathbf{D} = \mathbf{I} - \mathbf{R}^T $$ {#eq-d-matrix} The mixed boundary system is solved as: $$ \mathbf{M}\mathbf{j} = \mathbf{h} $$ {#eq-system} where $\mathbf{M}$ is assembled row-by-row from $\mathbf{C}$ (for known sources) and $\mathbf{D}$ (for known emissive powers), and $\mathbf{j}$ is the total radiant power vector. ## Advantages of the Anisotropic Extension 1. **Physical Fidelity**: Captures directional dependence of reflection-scattering, enabling more accurate modeling of real materials and surfaces. 2. **Analytical Framework**: Maintains the analytical structure of the exchange factor transformation, avoiding iterative Monte Carlo methods. 3. **Backward Compatibility**: Reduces exactly to the isotropic case when $\mathbf{\Gamma} = \mathbf{1}$, ensuring consistency with established methods. 4. **Preserved Properties**: Non-negativity and energy conservation are guaranteed (see chapter @sec-exchange_fact_anisotropic_extend for proofs). 5. **Flexibility**: Accommodates any anisotropic distribution within the constraint $0 \leq \mathbf{\Gamma}_{ij} \leq 1$, providing maximum modeling flexibility. ## Physical Applications For surfaces with specular components, media with anisotropic scattering phase functions (e.g., Mie scattering, Henyey-Greenstein, i.e. (ij,k)-three-index dependencies) and surfaces with oriented microstructure (e.g., brushed metal, fibrous materials), more complex models are required. ## Conclusion The anisotropic extension of the exchange factor transformation provides a mathematically rigorous and physically meaningful framework for modeling directional reflection-scattering in radiative transfer. By introducing the anisotropy matrix $\mathbf{\Gamma}$ and employing row-stochastic normalization, the method preserves all essential properties while significantly expanding the range of physical phenomena that can be accurately modeled.